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Can. J. Math. Vol. 47 (3), 1995 pp. 500-526

COMPARISON THEOREMS OF LIAPUNOV-RAZUMIKHIN TYPE FOR NFDE S WITH INFINITE DELAY

JOHN R. HADDOCK, SHIGUIRUAN, JIANHONG WU AND HUAXING XIA

ABSTRACT. Some comparison theorems of Liapunov-Razumikhin type are provided for uniform (asymptotic) stability and uniform (ultimate) boundedness of solutions to neutral functional differential equations with infinite delay with respect to a given phase space pair. Examples are given to illustrate how the comparison theorems and stability and boundedness of solutions depend on the choice(s) of phase space(s) and are related to asymptotic behavior of solutions to some difference and integral equations.

1. Introduction. In [19] and [20], Seifert provided informative examples to illustrate that Liapunov-Razumikhin type results related to (uniform) asymptotic stability for functional differential equations (FDEs) with finite delay do not always carry over readily to an infinite delay setting. Nevertheless, there have been some successful efforts to extend Liapunov-Razumikhin theory to retarded FDEs with infinite delay. Among these efforts are [4^-6], [8], [11-13], [15], [18] and [21].

For neutral functional differential equations (NFDEs)

(1.1) jD{Uxt)=f(Uxt\

the generalization of Liapunov-Razumikhin theorems becomes even more difficult, since the behavior of solutions depends heavily on the property of the so-called generalized difference equation

(1.2) D(t,xt) = h(t).

In the case where the delay is finite, under the restriction that the above D-operator is linear in the second argument and stable, Lopes [16] and [29] obtained some stability and boundedness results of Liapunov-Razumikhin type for finite delay NFDEs. In [25], one of the authors established some comparison principles of Liapunov-Razumikhin type for NFDEs with infinite delay on the space BC of bounded continuous functions with supremum norm.

The first author was partially supported by NSF under Grant No. DMS 9002431. The third author was partially supported by NSERC-Canada. Received by the editors November 23, 1993; revised March 24, 1994. AMS subject classification: Primary: 34K40; secondary: 34K20, 34D40. Key words and phrases: stability and boundedness, Liapunov-Razumikhin technique, NFDEs with infinite

delay, comparison method. © Canadian Mathematical Society 1995.

500

COMPARISON THEOREMS FOR NFDE S 501

As indicated in [7] among other places, there are certain drawbacks in restricting the setting to the space BC mentioned above. Along these lines one of the purpose of this paper is to establish stability and boundedness comparison theorems of Liapunov-Razumikhin type in a unified way so that the results are available both for general admissible phase spaces and the space BC of bounded continuous functions.

We obtain several comparison theorems of Liapunov-Razumikhin type for (uniform, asymptotic) stability and (uniform, ultimate) boundedness with respect to a given phase space pair (X, Y) with the intention that various choices of phase space pairs allow us to apply the obtained comparison theorems to study the existence of periodic solutions (see, cf. [28]) as well as the precompactness of positive orbits (see, cf. [7]) for NFDEs with infinite delay. The comparison theorems given here are reminiscent of those provided by Lakshmikantham and Leela [14] and Lopes [16] for finite delay retarded and neutral FDEs. However, it should be pointed out that the technicalities involved in dealing with neutral FDEs with infinite delay and the associated D-operator are by nature more complicated than those for retarded FDEs. Similarly, the difficulties are compound from the fact that we must be concerned also with the choice(s) of phase space(s) for infinite delay equations.

The rest of this paper is organized as follows. In Section 2, we introduce the definition and examples of a fundamental phase space which will be used throughout this paper. Section 3 is devoted to a precise description of stability and boundedness with respect to a given phase space pair. Several examples are also given to show that there is often a natural way to follow in connection with the choices involved in phase space selection. In Section 4, we prove several comparison theorems for uniform stability and uniform boundedness, and we show by examples how sufficient conditions of stability and boundedness depend on the choices of phase spaces. In Section 5, we use an argument of [26] to investigate asymptotic behaviors of the D-operator associated with a class of neutral integrodifferential and difference equations. Finally, in Section 6 we prove several comparison results for uniformly asymptotic stability and uniformly ultimate boundedness which include the classical Liapunov-Razumikhin theorems and a theorem in [23] as special cases.

2. Phase spaces and NFDEs with infinite delay. Let B be a linear space of Rn-valued functions on (—oo, 0] with a semi-norm/?(•) so that the quotient space B = B/p(-) of the equivalent classes of elements of B under the norm | • \B induced by /?(•) is a Banach space.

The space (B, | • |#) is called a fundamental phase space, if it satisfies the following conditions:

(Bl) there exists a constant £ > 0 such that |</>(0)| < KpQ») for all </> G B and (B2) for any t0 G R, S > 0 and any function x: (—oo, to + 8) —• Rn with xÎQ G B and

x: [to, to + 8] —> Rn continuous, we have xt G B for all / G [fy, to + Ô], where xt is defined by xt(s) = x(t + s) for s G (—oo, 0].

Among the typical and important fundamental phase spaces are the spaces Rn, Cr, Cg

and BC, which are defined below.

502 J. R. HADDOCK, S. RUAN, J. WU AND H. XIA

(a) Rn is the usual «-dimensional Euclidean space with the norm | • |. It can be regarded as a fundamental phase space where B is the linear space of all Rn -valued functions on (—oo,0] with the semi-norm/?(•) defined by /?((/>) = |</>(0)|.

(b) For any positive constant r > 0, Cr = C([—r,0],Rn) is defined as the space of continuous Rn-valued functions with the sup-norm | • |r defined by

|<£|r= sup 10(01, -r<s<0

where B is the linear space of all Rn-valued continuous fiinctions on (—oo,0] with the semi-norm p() defined by /?(</>) = |$[_r,o]|r.

(c) For any function g: (—oo, 0] —> [1, oo) satisfying the following condition: (gl) g: (—oo, 0] —> [1, oo) is a continuous nonincreasing function such that g(0) = 1. Let Cg denote the space of continuous functions which map (—oo, 0] into Rn such that

sup ——- < oo. s<0 g(s)

Then Cg equipped with the norm

5<0 g\S)

is a fundamental phase space. (d) BC is a special case of Cg with g(s) — 1 for all s G (—oo, 0]. For BC, the Cg norm

is denoted by | • |oo, that is

|0|oo = SUp|0(5)|. 5<0

Throughout this paper, we consider the following neutral functional differential equations with infinite delay

(2.1) JtD(t'Xt) =f(t'Xt)

subject to the following Cauchy initial condition

(2.2) xt0 = $

in which t > to > 0, B is a fundamental phase space, and D,f: [0, oo) x B —-» Rn. A solution of (2.1)—(2.2) (denoted by x(to, </>)(•)) is defined as a function x: (—oo, fo + #) —* Z?w, £ > 0, so that xto = </> G 5, x: [ o, o + 5).—>/?" is continuous, £)( ,x^) is differentiate on [ o, o + 6) and (2.1) is satisfied on [to, to + 6).

For fundamental results on existence, uniqueness, continuation and continuous dependence, we refer to [22] (for admissible phase spaces), [24] (for bounded continuous function spaces) and [27] (for general fundamental phase spaces).

We conclude this section by recalling some notation.

(e) For a continuous function V: [0, oo) x Rn —» [0, oo), the derivative of V along solutions of (2.1) is defined as

V{2.i)(t,D(t,xtj) = \imsup-\v(t + h,D(t + h,xt+h(t,xt))) - V(t,D(t,xt))],

where x: R —> 7?" is a solution of (2.1). (f) A wedge g is a continuous strictly increasing function Q: [0, oo) —» [0, oo) with

2(0) = o. (g) An unbounded pseudo wedge S is an unbounded strictly increasing continuous

function S: [0, oo) —• [0, oo).

3. Uniform stability and uniform boundedness: definitions and examples. Throughout the remainder of this paper, let X, Y and B be fundamental phase spaces with X C B and Y C B. We assume that D(t, 0) = /(*, 0) = 0 for all t > 0 so that equation (2.1)

possesses the zero solution. In this section, we introduce the concepts of stability and boundedness of neutral

equation (2.1) and its associated D-operator with respect to a phase space pair (X, Y). Examples are given to illustrate the dependence of these concepts on the choice of phase space.

DEFINITION 3.1. The zero solution of (2.1) is (X, Y)-uniformly stable if, for any e > 0 there exists a 8 > 0 such that [</> GX, \<f>\x <8 and / > to > 0] imply \xt(to, <J>)\Y <e.

DEFINITION 3.2. An operator/): [0, oo) x B —• Rn is (X, Y)-uniformly stable if there exists a wedge Q such that for any to E [0, oo), x:R —> Rn and e > 0 so that xto G X, x: [*o>oo) —-* Rn being continuous, \xtQ\x < Q{E) and sup,>,0 |Z)(f,.x,)| < Q(e), we have |jc,|y < E for all/ > t0.

DEFINITION 3.3. The solutions of (2.1 ) are (X, Y)-uniformly bounded if, for any a > 0 there exists a (3 > 0 such that [</> E X, \(j)\x < a and f > t0 > 0] imply |jc,(f0, </>)|y < /J.

Here, and in what follows, when discussing the boundedness of solutions to equation (2.1), we do not require that D(t, 0) = 0 and/(*, 0) = 0 for all t > 0.

DEFINITION 3.4. An operator Z):[0,oo) x B —* Rn is (X,Y)-uniformly bounded if there exists an unbounded pseudo wedge 5 such that for any to G [0, oo), x: R —•» /?" and / / > 0 so that xto G X, x: [/o,oo) —•* Rn being continuous, \xtQ\x < H and sup,>,0 |IKA*?)I < H, we have |jc,|y < S(H) for £ > t0.

REMARK 3.5. Usually it will be the case that either X = Y = B or X = B and Y = Rn, where Rn is regarded as an embedded subspace of constant functions in B. That is, if c G Rn, then the corresponding element in B is the constant function <j)c defined by (j)c(s) = c9s < 0. In this case \xt\y is merely \x{i)\.

Now we consider some examples for which the left-hand side of (2.1) has the form

d_ dt

JC(0 - V Btx{t - n) - / G(-s)x(t + s)ds- e(t) i=i J-°°

where e\ [0, oo) —* Rn is bounded, continuous and

M= sup \e(t)\. 0<Koo

In this case the operator D reads

(3.1) D(t, </>) = #>) - £ BM-rd - / G(-s)ct>(s)ds - e(t). i=\ J-°°

A special case, of course, is the "finite delay" left-hand side jt[x(t) — cx(t — r)], which often is found in the literature.

PROPOSITION 3.6. Suppose that Bt (i = 1,2,...) are n x n constant matrices,

G: [0, oo) —> Rnxn is continuous, {rt} is an increasing unbounded sequence of positive

numbers, and

oo ^o

(3.2) £ 1*1 + / lG(-*)l as = m < 1. i=i J-°°

Then for any m* G (m, 1), we can find a function g: (—oo, 0] —> [1, oo) satisfying (gl),

(gl) ^ ^lasu-^0+ uniformly for s G (-oo, 0], and

(g3) g(s) —> oo as s —> —oo

so f/iaf f/*e D-operator defined by (3.1) is (Cg, Rn)-uniformly bounded with S(H) = yr^r-

Moreover, ife = 0, r/ze« the D-operator defined by (3.1) is (Cg,Rn)-uniformly stable with

Q(e) = (l-m*)e.

PROOF. In [7], it is shown that (3.2) implies the existence of a function g: (—oo, 0] —» [1, oo) satisfying (gl), (g2) and (g3) such that

oo ^

(3.3) J2\Bi\g(-n)+J \G(-s)\g(s)ds<m\ i=\ J-°°

For any *o £ [0,oo), x:R -^ Rn and H > 0 with JC,0 G Cg, x: [*o,oo) —> i?w being continuous, \xto\cg < Hand sup,>,0 |D(f,jC/)| < H, let

oo ft

h(t) = x(t) - V Btx(t - n) - / G(t - S)JC(S) & - <?(*)•

If there exists a r > fo such that |JC(T)| = maxto<s<T \x(s)\, then

\x(f)\ = V 5/jc(r - r,-) + / G(-s)x(s) ds + e(r) + A(r)' l ;_1 J—CO '/=!

Let K be an integer so that rK < r — to < rjc+\. We have

K T ) | < E \Bi*(T-n)\ + E \Bix«0 +T-I0- r,)\ 1=1 i=K+l

+ ['" IG(T- s)\x(s)ds+ TIG(T - S)|JC(S)ds + H+M J—OQ J to

<[EIAI+£|GC--S)|&]W)I

'+ £ i^i^)g<T-'07'>k/T-,'0-tl

i=K+i g(-n) gfr-to-n) + f \G(r -to- lOlgfo + « - r) . fM) . k / f d» + / / + M

J-00 g ( ? 0 + M - T ) g(w)

<[X>l+ / ° |G(-5)| ]|x(r)|

< [ E \Bi\g(-n) + f |G(-s)|g(s)<fc] max{|x(r)|, K | C J + / / +Af L*=i J-°° J

<m*max{ |x ( r ) | ,K |cJ+ / / + M

by (3.3), since g(s) > 1 for all s G (—00,0] by (gl). Now, if \x(r)\ > \xto\cg, it follows from the above inequality that

|JC(T)| <m*|jc(r)|+// + M,

which implies

Therefore

WT)| < H + M

1 — m*

H + M\ H + M x(r) < max x/n \r, = . 1 w i - y t0\Lg, x_nfj i-m*

This proves the (Cg, i?rt)-uniform boundedness of D. Similarly, we can prove the (Cg, R

n)-uniform stability of D when e = 0. •

Following the proof of Proposition 3.6, one can show the following

PROPOSITION 3.7. Under the assumptions of Proposition 3.6, the D-operator de-\edby (3.1) is (BC, Rn)-uniformly bound

stable when e = 0 with Q{e) = (1 — m)e. fined by (3.1) is (BC,Rn)-uniformly boundedwith S(H) = ^ andis(BC,Rn)-uniformly

The next result allows more flexible Bt's and G and includes the previous propositions. For the sake of brevity, the proof is omitted.

PROPOSITION 3.8. Suppose that Bt: [0, oo) x Rn —> Rn, G: [0, oo) x R x Rn —> Rn

and e\ [0, oo) —• Rn are continuous. Suppose also that there exist constants K > 0 and k\ G [0,1) such that

(i) for any t>to>0 and <\> G X, we have

E sup \Bi(t,<Ksj)I + / \G(t,to + s,<t>(s))\ds <K\4>\x;

(ii) for any continuous x: [to, oo) —» Rn, we have

N rt

YJ\Bi(t,x(t-ri))\+ \G(t9s9x(sj)\ds<ki sup \x(s% i = l Jto t0<s<t

where N is an integer so that r^ < t — to < r^+\. Then the D-operator

oo «o

D(U </>) = <K0) ~ E M'* ^(- r ' ) ) " / G('> ' + *> ^ ) ) * " e® i=\ J-°°

is (X,Rn)-uniformly bounded with S(H) = ( K j ^ + M if sup,>0 \e(t)\ < M < oo, a/irf is

(X,Rn)-uniformly stable with Q(e) = j^e ife = 0.

We conclude this section with a simple result to contrast with the finite delay equations.

PROPOSITION 3.9. If there exist constants KUK2 > 0 such that for t0 G [0,oo), x:R-^ Rn with xto G X, x: [to, oo) —> Rn being continuous and

\xt\Y<Kl\xt0\x + K2 sup \D(t,xt)\. t0<s<t

Then the operator D is (X, Y)-uniformly stable and(X, Y)-uniformly bounded with Q(e) = j ^ — andS(H) = (K\ + K2)H, respectively

It follows that if D: [0,oo) x Cr —> Rn is stable in the sense of Lopes [16] (or [9]), then this D-operator is (Cr, R

n) (or (Cr, Cr))-uniformly bounded and (Cr, Rn) (or (Cr, Cr))-

uniformly stable.

4. Uniform stability and uniform boundedness: comparison theorems. In this section, we are concerned with comparison theorems and their applications to uniform stability and uniform boundedness. We start with a uniform boundedness theorem.

THEOREM 4.1. Suppose the operator D is (X, Rn)-uniformly bounded and there exist unbounded pseudo wedges Wi{i = 1,2,3), a constant M > 0 and continuous functions V: [0, oo) x Rn —• [0, oo), W\ [0, oo) x [0, oo) —> [0, oo) such that

(i) \D(t^)\<W3(\$\x); (ii) m(\x\)<V(t,x)<W2(\x\);

(Hi) for any (to, </>) G [0, oo) xXandanyx: R-* Rn with xto = </> andx: [to, oo) —> R" continuous, we have

(4.1) Vv.i){t9D(t9xtj) < w(t, V{t,D(t,xt)))

att > to where max{\xtQ\x, suPt0<s<t\x(s)\} — SoW\x oV(t,D(t,x$) andV(t,D(t,xt)) > M, here S is given in Definition 3.4;

(iv) the solutions ofz = W(t,z) are uniformly bounded. Then the solutions of (2.1) are (X, Rn)-uniformly bounded.

PROOF. For any (t0, </>) G [0, oo) x Xand a > 0, if \<))\x < OL, then

\D(toM < W k ) < W3(a)

and

V(t0,D(to,<t>)) < W2{\D(toA)\) <W2o W3(a).

Choose 7 = max{a,M, W\(oc), W2 o W3(a\ W\ o S~l(a)}, by assumption (iv), there exists a f3\ > 0 such that for any to > 0, if z(t) is the maximal solution of

z(t) = W(t,z(t)) > *0b) = %

then |z(0| < ft for all t > t0. We claim that

(4.2) V(t,D(t,xt))<z(t)

for all t > to. Suppose it is not true, then we can find a positive integer m and a real number t\ > to such that

V(s,D(s,xs))<zm(s)

fors G [to,t\X and V(tuD(ti9xtl))=zm(ti),

and there exists a sequence rn —+ /J such that

F ^ D ^ X r , , ) ) >zw(r„)

for « = 1,2,..., where zm(t) is a solution of the initial value problem

\zm(t) = vr{t,zm{t)) + ± \ Zm(to) = 7

(see, c/ [14]). Therefore

. , v ^ ^ ( r ^ x ^ ) ) - ^ ! , ^ , ^ ) ) n*i,D(f!,*,,))•> hm - ^ *——^ ^

x ' n-^oo rn — t\ > l i m Zm(T„) ~ Zm(t{)

~ n-^oo Tn — t\

= W(tuzm{ti)) + -v ' m

= w(tl,V(ti,D(tl,xllj)) + ^.

On the other hand, since zm{i) is increasing, it follows that

Wx(\D(s9xs)\) < V(s9D(s9xs)) <zm(s)<zm(ti)

for to <s <t\, that is

\D(s9xs)\<wxi(zm(tx))

for t0 < s <t\. Since 7 > Wx(a\ we have

|</>U < oc < Wx\l) = Wx\zm(to)) < Wx\zm(tx)).

By the Definition 3.4, we have

Hs)\<sowxx(zm(txj)

for to < s < t\. Moreover, the choice of 7 gives

\4>\x <a<So Wxx(l) < So W^{zm{tx)).

Hence (4.1) implies that

V(tuD(tuxtl)) < w(tuV(tuD(tuxt]))),

which contradicts (4.3). Therefore, (4.2) must hold. Thus we have that for all t > to

V(t,D(t,xt))<z(t)<(3u

which implies

\D(t,xt)\ < tf^GSi)

for all t > t0. Since 7 < /3i, and hence \<j>\x < a < W^x{j5x\ by the (X9Rn)-m\\ïorm

boundedness of the operator/), we have

\x(t)\ 0 and continuous functions V: [0, oo) x Rn —-» [0, oo), W\ [0, oo) x [0, oo) —> [0, oo) such that

(i) \D(tM<m<t>\x); (n) Wx{\x\)<V(t9x)<W2(\x\); (Hi) for any (to, </>) € [0, oo) x X and any x:R—>Rn with xtQ = (j> andx: [to, oo) —•» Rn

continuous, we have

V{2A)(t9D(t9xt)) < w(t9V(t9D{t9xt)))

att>to where max{\xto\x, sup,o<5<, \x(s)\} < Q~x o W\x o v(t9D(t9xt)), here Q is given in Definition 3.2;

(iv) the zero solution ofz = W(t,z) is uniformly stable. Then the zero solution of (2.1) is (X, Rn)-uniformly stable.

The above theorems allow us to determine the uniform boundedness and uniform stability for NFDEs with infinite delay. In the following example, we examine (BC9R

n)~ and (Cg, iT)-uniform boundedness (stability) of certain integrodifferential equations. In the Cg case we see how the choice of underlying phase space is involved in determining the uniform boundedness and uniform stability.

EXAMPLE 4.3. Consider the following linear nonhomogeneous Volterra integrodifferential equation

dt

°0 rt 1 x(t) - TBtxit - n) - G(t- s)x(s) ds -f(t)

,-=i J-°° -I oo rt

= Ax(t) + V Atx(t - n) + H(t- s)x(s) ds + k(t). : 1 J — OO

(4.4) = Ax(f) 4

/=1

We assume that (1) A is a stable n x n constant matrix. That is, there exist an n x n positive definite

matrix P and constants (3 > a > 0 such that

ATp + PA = -I, a1xTx < xTPx < (32xTx;

(2) / , k: (—oo, +oo) —-»• Rn are continuous and \f(t)\ < M\, \k(t)\ < Mj for some constantsM\,AÎ2 > 0;

(3) Z°Zl\Bi\+f-o0\G(-s)\ds = m<l; (4) E£ , | 4 |+£ 0 0 | t f ( - . s ) |< fe<oo . Let V(x) — xTPx. Then for the D-operator defined by

D(t, 0) = </>(0) - g BM-n) - [° G(t- sMs) ds -f(t\

we have

{oo

AD{t,xt) + Y^iAi + ABt)x{t - n)

+ J' [H(t -s)+AG(t-s)]x(s)ds + k{i) +Af(t)\

!

oo AD(t,xt) + YtAi +ABi)x(t - n)

+ f* [H(t-s)+AG(t-s)]x(s)ds + k(t)+Af(i)\ PD(t,xt)

(oo

YtAt+ABiW-n)

+ f_JH{t -s)+AG(t- s)]x(s) ds + k(t) + Af(t)\.

By Proposition 3.7, D is (BQ^-uniformly bounded with S(H) = ^ L . If

\x(s)\<S([DT(t,xt)PD(t,xt)]ï/a)

for s < t, then

|x(5)l —rrs— for 5 < f, and therefore

r CO

F(4.4)(f, £>(*,*,)) < -DT(t,x,)D(t,Xl) + 2|Z)(?,x,)| |/»| E \Ai+ABi\

+ / #( / - s) + AG(t - s)\ ds\ " ' ' J — OO J

\ — m

where

+ 2\D(t,Xl)\\P\\k(t)+Af(t)\

Wl = V \At +ABt\ + / | / /(-s) + AG(-s)\ ds.

; _ 1 ^ — C O Thus, by Theorem 4.1 we have

PROPOSITION 4.4. Under the conditions (l)-(4), if

(0 «i < %3* . "• (^ wi < ^ p f andf(t) = k(t) = Ofor t > 0,

then the solutions of (4.4) are (BC,Rn)-uniformly bounded.

REMARK 4.5. Using Theorem 4.2, we can prove that if assumption (ii) in Proposition 4.4 holds, then the zero solution of (4.4) is (BC, Rn)-xmiform\y stable.

We now consider the (Cg, /?w)-uniform boundedness and uniform stability of (4.4). We will see how the choice of phase space enters into the considerations. Let m < m* < 1. By aresult in [7], there exists a continuous functiong: (—oo, 0] —• [1, oo) satisfying (gl), (g2) and (g3) such that

and

E \Bi\g(-rd + f° \G(-s)\g(s)ds < m* /=i J-°°

£ \Ai\g(-n) + / \H(-s)\g(s)ds < oo. • 1 J— CO

By Proposition 3.6, D is (Cg, -uniformly bounded with

1 — AW*

If

and [DT(t,xt)PD{hxt)]

I ^ l (lDT(t,x,)PD(t,xt)]^tA 8 1 — m* V a J

| A r ( 5 ) l ^r^i « +MlJ for f0 < s < t, then

r OO

V(4A)(t,D(t,xt)) < ~DT(t,xt)D(t,xt) + 2\D(t,Xl)\ \P\ £M,+AB i \g{-r i )

+ 1° \H(-S)+AG(-s)\g(s)ds\—^\^\D(t,xl)\+Ml} j-oo j i — m lex J

+ 2|fl(/,x/)||/,|(A/2+M|ilf1)

V (1 —m*)aj

+ AP\(^h +M2 + \A\Mx)\D(t,xt% VI — rrr J

where

U2 = £ M/ + ^ : | g ( - r f ) + / | / /(-s) + AG(-s)\g(s)ds.

Therefore, by Theorem 4.1 we have

PROPOSITION 4.6. if

0 U2 < ° g ^ , or W u2 < ( 1 2 ^ £ ««<//(*) = Kt) = 0/or / > 0,

//ze« £/ze solutions of (4.4) are (Cg,Rn)-uniformly bounded.

REMARK 4.7. Using Theorem 4.2, we can prove that if assumption (ii) in Proposition 4.6 holds, then the zero solution of (4.4) is (Cg,i?

w)-uniformly stable. It is interesting to note that the (Cg, R

n)-unifovm boundedness (stability) under the conditions in Proposition 4.6 implies the (#C,i?w)-uniform boundedness (stability) by Proposition 4.5.

5. Uniform asymptotic stability and uniform ultimate boundedness: definitions and examples. In this section, we introduce the concepts and sufficient conditions of asymptotic stability and ultimate boundedness of neutral equation (2.1) and its associated D-operator with respect to a given phase space pair (X, Y).

DEFINITION 5.1. The origin (X, Y)-attracts the solutions of (2.1) uniformly if, for any M > 0 and rj > 0 there exists a T(r], M)>0 such that for any solution *(/) of (2.1M2-2) defined for t > to with max{|x,0|x, sup5>,0 \x(s)\} < M, we have |x,(/o,</0|r < V f° r

t>to + T(r]9M).

DEFINITION 5.2. The solutions of (2.1) are (X, Y)-weakly uniformly ultimately bounded for bound B > 0 if, for any /3 > 0 there exists a T(j3) > 0 such that for any solution x(0 of (2.1 )-(2-2) defined fort > t0 withmax{|x,0|x, sup5>,0 \x(s)\} < /3we have \xt(t0, <j>)\Y tQ + T(/3).

DEFINITION 5.3. Suppose that D(t, 0) = f(t, 0) = 0. The zero solution of (2.1) is (X, Y)-uniformly asymptotically stable if, it is (X, Y)-uniformly stable and there exists a constant So > 0 such that for any e > 0 there is a T{e) > 0 so that for any solution x(t) of (2.1H2.2) defined for / > t0 with \xto\x < 6Q, we have \xt(t09 4>)\y < e for t > t0 + T(e).

DEFINITION 5.4. The solutions of (2.1) are (X, Y)-uniformly ultimately bounded for bound B > 0, if for any a > 0 there exists a T(a) > 0 such that for any solution x(t) of (2.1H2.2) defined for t > t0 with \xto\x < a, we have |jc,(fo,</0|y to + T(a).

Obviously, if D(/,0) = f(t,0) = 0, then the (X,fln)-uniform stability of the zero solution of (2.1) and the (X,/^-uniformly attractivity of the origin imply the (X,Rn)-uniform asymptotic stability of the zero solution of (2.1). Similarly, if the solutions of (2.1) are (X,/^-uniformly bounded and (X, jRw)-weakly uniformly ultimately bounded, then the solutions of (2.1) are (X, /^-uniformly ultimately bounded.

DEFINITION 5.5. An operator D is called (X, Y)-pseudo uniformly asymptotically stable if there exists a wedge P such that for any e,M > 0 there is a T\(e,M) > 0 such that for any to G [0,oo), x:R —> Rn with xto £ X, x: [to,oo) —» Rn being continuous, max{|xj;r, sup5>,0 |x(s)|} < M and sup5>,0 |D(5,^)| < P(e)9 we have |x,|r < e for />^o + ri(e,M).

An operator D which is both (X, y)-uniformly stable and (X, 7)-pseudo uniformly asymptotically stable is called (X, Y)-uniformly asymptotically stable.

DEFINITION 5.6. An operator D is called (X, Y)-pseudo uniformly ultimately bounded if there exists an unbounded pseudo wedge B such that for any M\, M2 > 0 there is a r2(Mi ,M2) > 0 such that for any t0 G [0,00), x:R-^Rn with xto e X, x: [f0,00) —> /?* being continuous, max{|x/o|^, sup5>/o |x(.s)|} < Afi and sup5>/o |D(s,jc,s)| < M2, we have \xt\Y< B(M2) for t > To + r2(Mi,M2).

An operator Z) which is both (X, y)-uniformly bounded and (X, Y)-pseudo uniformly ultimately bounded is called (X, Y)-uniformly ultimately bounded.

To illustrate the above concepts, we consider the D-operator defined by

(5.1) D(t,<l>) = m-Y,Bi(t><l>(-rij)- / G(t9t + u9<l>(u))du9

where £,(/ =1 ,2 , . . . ) : [0,00) x Rn —• Rn and G: [0,00) x R x 7?" -» Rn are continuous, {r,-} is an unbounded increasing sequence of positive real numbers. By a similar argument to that in [23], we obtain the following

PROPOSITION 5.7. Suppose that there exists a nonnegative constant 1 < 1, and for any e > 0, M > 0 there exists an integer K = K(e, M) > 0 such that for any x:R-^ Rn

with xto E X, x: [to, oo) —> Rn being continuous andm&x{\xtQ\x, sup5>,0 |x(s)|} < M, we

r*\G(tJ + s9xt{s))\ds+ £ I ^ ^ J c K - r / ^ k e

J_r |G(/,/ + x ^ ) ) | & + E | ^ ( ^ ^ ( - r / ) ) | </ f_maxjx(s) |

yor / > to+rK- Then the D-operator defined by (5.1) is (X,Rn)-pseudo uniformly asymptotically stable with P(r) = qr, where q is any given constant in (0,1 — /).

PROOF. Choose S > 0 so that / + q + S < 1. Let K = K(6e, M),x:R-^ Rn be given with xto E X, x: [to, oo) —• i?w being continuous and satisfy the following inequalities

max{ |jc/()\x9 sup \x(t)\} < M, sup \h(s)\ < qe, s>t0 s>t0

where

h{i) = x(t) - Y,Bi(t,x(t - nj) - J^ G(t,s,x(s)) ds.

Then for any t > to + r^, we have

and

/ rK\G(t,s,x(s))\ds+ £ {Bihxtt-njSlKSe

JlrK\G(t,s,x{s))\^

Consider now the consecutive intervals /„ = [to + nrx, to + (n + l)r#] for n > 1 and find tn E [to + nrK, fa + (n + 1)^] so that |x(^)| = max5G/w |*0)|. Then

\G(tn,s,x(s))\ds+ J2 mtn,x(tn-n))\ ~°° i=K+\

+ f^ \G(tn,s,x(s))\ + Y\Bi{tn,x(tn - n))\ + \h(tn)\

<bs + qe + l max \x{s)\.

Therefore either |*fti)| <(q+S)e + l\x(tn-i)\

if there exists f E [/„ — rK, fa + nrK] so that |jc(f*)| = TRa*tH-rK<s<tn \x(s)\, or

\x(tn)\ < (^+«)e+ 71^)1

if no such f* exists. So we assert that either

(1) |x(^)| < ^ e < e, for k > N, where N is some integer, or

(2) |x(/„)| <{q + 5)e + l\x(tn.x % for n = 2 , 3 , . . . .

In the second case, we have

\x(tn)\ < (q+S)e(l + / + /2 + • • • + f"3) + l"~2\x(t2)\

for n — 2,3, Choose TV* so that q + b l n ( l - | — ) £ - l n M /In/ . N * > 2 +

Then for n > TV*, we have

i , M q+à (, q+à\ \Atn)\ < \=)e + v" i ^ 7 r = e'

This shows that |JC(/)| < e for all t> to+N*rx. This completes the proof. •

Similarly, we have the following

PROPOSITION 5.8. Suppose that there exist constants C\, C2 > 0 and I £ [0,1 ) such that for any M\ > 0 there exists an integer K = K(M\ ) so that for any x.R —> Rn with xto £ X, x: [to, oo) —» Rn being continuous and max{\xto\x, sup5>,0 |JC(S)|} < M\, we have

\G(t,t + s,xt(s))\ds + Y, \Bi(tM-nj)\<Ci -°° i=K+\

and

\G(U t + s,xt(s))I ds + Y\Bi(t,x(t - nj)I < / max |JC(^)| + C2 rK

] X n /=1 t-rK<s<t

for t >to + rK, then for any bounded continuous f: R —> Rn the D- operator

D(t, <f>) = <H0) - £ s<(' ' <K-rtj) - J^ G(t, t + u, <Kuj) du -fit)

is (X,Rn)-pseudo uniformly ultimately bounded with

B(M2) = C i + C 2 + M 2 + S u p ^ ^ ( 0 l ^

q

where q is any given constant in (0,1 — /).

We complete this section by presenting a simple result to contrast with the finite delay case (cf. [3]).

PROPOSITION 5.9. If there exist constants K\, K2 and a > 0 such that for any to G [0, oo), x:R-^ Rn with xto G X, x: [to, oo) —> Rn being continuous, we have

\xt\Y<Kxe-a^\xtQ\x + K2 sup \D(s9xs)\ t0<s<t

for t > to, then the D-operator is (X9 Y)-uniformly asymptotically stable and (X, Y)-uniformly ultimately bounded with P(e) = as and B(M2) = M2/a, respectively, where a is any given constant with 0 < a < 1 / K2.

REMARK 5.10. By Proposition 5.9, the stable D-operator introduced by Cruz-Hale [3] for NFDEs with finite delay is (Cr,R

n) (or (Cr, Cr))-uniformly asymptotically stable and uniformly ultimately bounded. Moreover, Melvin [ 17] showed that even for NFDEs with finite delay, the result obtained in Proposition 5.7 and 5.8 is very sharp in the sense that this is the best possible result such that the D-operator retains its stability under appropriate perturbation on rz. For details we refer to [17].

6. Uniform asymptotic stability and uniform ultimate boundedness: comparison theorems. In this section, we provide a general result and several corollaries for uniform asymptotic stability and uniform ultimate boundedness of neutral equations with infinite delay.

To present an as general as possible comparison theorem, we introduce the following concept.

DEFINITION 6.1. Let W\ [0, oo) x [0, oo) —* R be continuous. The solutions of

(6.1) z=W(t,z9S)

are strongly uniformly asymptotically convergent to zero, if (1) for any 6,rj9M > 0 there exists S\(69r]9M) > 0 such that for any nonnegative

solutionz(0of(6.1)through(r0,z0) G [0,oo)x[0,M],wecanfindrG [to,t0+S{(69r]9M)] so that z(r) < r/;

(2) for any b9a9M > 0 there exists S2(S9a9M) > 0 such that for any nonnegative solutionz(r)of(6.1)through(t09z0) G [S2(69a9M)9oo) x [0,M],wehavez(f) <z(fo) + cr for t > to.

EXAMPLE 6.2. If W(t9 z, 8) = - W(z) +g(t)9 where W\ [0, oo) -> [0, oo),g: [0, oo) -> R are continuous, W{x) > 0 for all x > 0 and J™g(s)ds < +oo, then the solutions of (6.1) are strongly uniformly asymptotically convergent to zero.

THEOREM 6.3. Suppose that the operator D is (X9 Rn)-pseudo uniformly asymptotically stable, and that there exist wedges W((i = 1,2,3), continuous functions V: [0, oo) x Rn —• [0, oo) and W\ [0, oo) x [0, oo) x [0, oo) —> R such that

(i) for any x:R-^> Rn with xto G Xandx: [to, oo) —-> Rn being continuous, we have

\D(t9xt)\ < W^(max{|*,0|jr, sup |*00|}), t > t0; t0<s<t

(ii) Wl(\x\)<V(t,x)<W2(\x\); (Hi) for any M > 0 and b > a > 0 there exist 8 > 0 and h > 0 such that for any

x:R —> Rn with xto G X andx: [to,oo) —> 7?" Z>e/«g continuous, at any t > t0 + h with max{|x,0U, sup,o<,<, \x(s)\} < M9a < V(t9D(t9xt)) < b and V(s,x(sj) < ^ o P~l o

W\x (v(t9D(t9xt))) + <S for s e[t- h9t\ we have

V{2A)(t9D(t9xt)) < w(t9V{t9D{t9xt))98)9

where P is given in Definition 5.5; (iv) the solutions of (6.1) are strongly uniformly asymptotically convergent to zero. Then the origin (X, Rn)-attracts solutions of (2.1) uniformly.

PROOF. Let M, 77 > 0 be given and x{i) be a solution of (2.1 ) defined for t > to with

max{ |*,0 \X9 sup \x(s)\ ) < M. s>t0

Then V(t,D(t,xtj)<W2oW3(M)

for t > to. Choose h = h(r\9M) and 8 = 8{r\9M) > 0 such that if for some t > to + h, we have

y ; < V(t9D(t9xtj) < W2 o W3(M)

and V(s9x(s)) < W2oP-{ o W^x (y(t9D(t9Xtj)) + 8

fox s E [f — h9t]9 then

V{2A)(t9D(t9xt)) < w(t9V(t9D(t9xt))98).

For the 8 > 0 chosen above, find a = a(r\9M) > 0 and (3 = (3(j]9M) > 0 so that (3 < ^ f ^ and W2oP~x o W{\s+u)-W2oP-x o W^\s) < 8 for ^USaE < 5 < W2oW3(M) and 0 < w < a + 2/3. Let N(j]9M) be a positive integer such that

W\ (P(n)) + N(a + j3)>W2o W3(M)9

and let ei=Wx(P(TiJ)+Ka + l3)

for 1 fo, we have

V(t,D(t9xt)) <W2o W3(M) < eN.

By the (X9 Rn)-psexxdo uniformly asymptotic stability, this implies that

\x(t)\<p-loWi\£N)

for t > to + T$/>-' o WYl(eN),M).

If JV > 1 and V(t,D(t,xt)) > eN^ - j3 for all t > t\, where

t\ = t0 + T,(F"1 o W^\eN),M) +h+S2

Si = S2{b(rl,M),(i(7],M), W2 o W3(M)),

and

then we have

(6.2) V2

K"> < V(t,D(t,xt)) <W2o W3{M)

fort > t*{,and

V{s,x(sj) < W2(\x(s)$

<W2oP~] oW^(eN)

<W2oP~x oWYl(v(t,D(t,xt)))

+ W2o P~{ o W^\eN) -W2o Pl o W^ (v(t,D(t,x,)))

< W2oP~{ o W\x (v(t,D(t,xt))) +S

for s G [t — h, t] and t > t*v This implies

V(t,D(t,xtj) < W(t, V(t,D{t,xt)),è)

for t>t\, and thus by the well-known comparison principle (see, cf. [14]), we have

V(t,D{t,xt))<z{t;t\,W2oWi{M))

for t > 4,wherez(t; fx, W2 o W3(MJ) is the solution of (6.1) through (t*u W2 o W3(MJ). By assumption (iv) there exists t\* G \t\, t\ + S\ ] such that

s Wi(P(i])) z{t\*;ti,W2oW3(M))< l\K">,

where

This implies

( W{(P(r})) \ 5, = Si kr?,M), y • Wl ° W^M)) •

, W, (P(î]j) V{tt,D{t\\xrcj) < —K - U ^ 2

which is contrary to (6.2). Therefore there must be a T G [t\, t* + Si] such that

V(r,D(T,Xr))<eN-i-p.

If there exists a T* > T SO that F(T*,D(T*,JCT*)) > £#_i, then there must bear** G [T9T*]

such that

F(T**,D(T**,XT**)) = eN.x - (3 < V(t9D(t9xt))

for t G [T**,T*). Using the same argument as above, we can prove that

V{t9D{t9xt)) < w(t9V(t9D(Lxt))9S)

for / G [r**,r*), thus by assumption (iv) we have

V(T\D(T\XT*)) < Z(T*;T**9 F(T**,Z)(T**,XT**)))

< F(T**,D(T**,XT**)) + /3fo,M) = eN-X.

This contradicts to K(r*,D(r*,xT*)) > £#-i- Therefore

V(t9D(t9xt))<£N-i

holds for all t > r, hence for all t > t0 + Tx(P~x o W\x(eN)9M) +h + S2 + S{. Following a similar argument, we can prove that

v(t9D(t9xtj) <£N-k

for t > t0 + 7 (r/, M), where

7 7 ( r / , A ^ ) - è ^ ( ^ " l o ^ r 1 ( ^ - / + l ) , M ) + ^ + iS2+^1]. 1=1

Thus V(t9D(t9xt))<eo = Wl(P(riJ)

for f > t0 + T*N(l,M). It follows that |jc(f)| < 1 for t > t0 + ^(77,M) + TX{T]9M). This completes the proof. •

Likewise, we can prove the following

THEOREM 6.4. Suppose that the D-operator is (X9 Rn)-pseudo uniformly ultimately bounded, and that there exists a constant M > 0, unboundedpseudo wedges Wt(i = 1,2,3), continuous functions F:[0,oo) x Rn —> [0,oo) and JT:[0,oo) x [0,oo) —> R such that

(i) for any x:R—^Rn with xto G Xandx: [to, 00) —-> Rn being continuous, we have

\D(t9xt)\ < W3(max{\xto\x, sup \x(s)\))9 t > t0;

(H) Wx{\x\) < V(t,x) < W2{\x\); (Hi) for any M\ > 0 and b > M there exist 8 > 0 and h > 0 such that for any

x:R-^Rn with xto G X andx: [to, 00) —> 7?" frewg continuous, at any t > to + /* w/Y/z

max{|jt/o|;r, suptQ<s<t \x(s)\} < M\, a < V{t9D(t9xt)) < b and V(s,x(s)) < W2 o B o

W\x (v[t9D(t9xt))) +6 for s <E[t- A,/], we have

V{2A)(t9D(t9xt)) < w(t9V(t9D(t9xt))9è)9

where B is given in Definition 5.6; (iv) the solutions of (6.1) are strongly uniformly ultimately bounded. Then solutions of (2.1) are (X9R

n)-weakly uniformly ultimately bounded.

Here, by strongly uniformly ultimate boundedness of solutions of (6.1), we mean that there exist constants M* > 0 and AT* > 0 such that

( 1 ) for any 8 > 0 and M>M* there exists S3 (£, M) > 0 such that for any nonnegative solution z(t) of (6.1) through (to,z0) G [0,oo) x [M*9M]9 we can find a r G [to, to + S3(69M)] so that Z(T) < AT*;

(2) for any cr, M > 0 there exists S4 {a, M) > 0 such that for any nonnegative solution z(0of(6.1)through(t0,z0) G [S4(a,M),oo)x[M*,M],wehavez(0 <z(t0)+crfort > t0.

We now present some utilizable corollaries of Theorem 6.3 and 6.4. First we notice that Theorem 6.3 contains the classical Liapunov-Razumikhin type theorem.

THEOREM 6.5. Suppose that the operator D is (X9Rn)-pseudo uniformly asymp

totically stable, and that there exist wedges Wt(i = 1,2,3,4), continuous functions V: [0,00) x Rn —> [0,00) andq: [0,00) —» [0,00] with q(s) > s for s > 0, such that

(i) for any x\R^> Rn with xto G Xandx: [to, 00) —» Rn continuous, we have

\D{t9xt)\ < W3(max{\xt0\x, sup \x(s)\})9 t > t0; t0<s<t

(U) Wi(\x\) < V(t9x) < W2{\x\); (Hi) for any M > 0 and b > a > 0 there exists h > 0 such that for any x: R —» Rn

with xtQ G X and x:[to,oo) —> Rn being continuous, at any t > to + h with max{|jt/o|x, sup/o<s<, ^(s)!} < M9a < V(t9D{t9xt)) a > 0, define

6 = inf{qo W2 oP~l o W^(s)- W2oP~x o W\l(s);a <s<b}>0.

Obviously, if a < V(t,D(t9xt)) < b and if

V(s9x(sj) < W2oP~l o W\x (v{t9D{t9x^)) +S

for s e[t — h,t], then

V(s9x(sj) <qoW2op-lo W^x (v(t,D(t,xt)))

and thus

V{2A)(t,D(t,xt)) < -W4(\D(t,xt)\) < -W4 o W2l (v(t,D(t,xt))).

Therefore the origin (X,/T)-attracts solutions of (2.1) uniformly by Theorem 6.3. •

The following theorem, also as a consequence of Theorem 6.3, contains a variant of

the main result in [23].

THEOREM 6.6. Suppose that the operator D is (X, Rn)-pseudo uniformly asymp

totically stable and that there exist wedges Wi(i — 1,2,..., 5), continuous functions

V: [0, co) x Rn —> [0, oo), F: [0, oo) x [0, oo) x [0, oo) -> R, k: [0, oo) —> [0, oo) with

$° k(f) dt < oo such that

(i) for any x\R—*Rn with xto G Xandx: [to, oo) —• Rn being continuous, we have

\D(t,xt)\ < ^3(max{|x,0|, sup \x(s)\})9 t > t0; to<s<t

(ii) Wx(\x\) < V(t9x) < W2(\x\);

(Hi) F(t, V, W2 o P~l o Wï\V)) < -WA{V);

(iv) \F(t, V,NX) - F(t, V,N2)\ < W5(\N{ - N2\) + k(t)\Nx - N2\ for t > 0, V >

09NuN2 > 0 ;

(v) for any a > 0 and M > 0, there exists h > 0 such that for any N > 0 and any x:R —> Rn with xto E X andx: [to, oo) —-» Rn being continuous, at any t > to + h with max{\xtQ\X, s\xpto<s<t \x(s)\} < M and swp_h<s<t V\s,x(s)) < N, we have

V(2A)(t,D(t,xt)) < F(t9 V(t9D(f9xtj)9N) + a.

Then the origin (X, Rn)-attracts the solutions of (2.1) uniformly.

PROOF. For any b > a > 0, choose positive constants 8 = W^x{jW4{a)^j and a = \ W4{a). Then for any x: R —» Rn withx,0 G Xandx: [to, oo) —> Rn being continuous, at any t>t0 + h with max{|x,0|X, sup,o<5<r \x(s)\} <M,a< F(7, £>(/, x,)) < b and

maX< V(s9x(s)) < W2 o P~{ o W\x (v(t,D(t,xt))) +8,

we have

V{2A)(t9D(t9xt)) < FU V(t9D{t9xt))9 W2oP~x o W\x (v{t9D(t9x$)) +«) + °

< p\t9 V(t,D(t9xt))9 W2oP'1 o W\x (v(t9D(t9xt)))]

+ F^t9 V{t9D{t9xt))9 W2 o r 1 o W~xx (v(t9D(t9xt))) + <$}

- FU V(t9D(t9xt))9 W2 o P~x o Wxx (v(t9D(t9xt))j\ + a

<-W4 (v(t9D(t9xt))) + W5(S) + a + k(t)5

<-\w4(v(t9D(t9xt)))+k(t)8.

Therefore the origin (X, #w)-attracts the solutions of (2.1) uniformly by Example 6.2 and Theorem 6.3. •

Likewise, by Theorem 6.4, we can prove that

THEOREM 6.7. Suppose that the D-operator is (X, Rn)-pseudo uniformly ultimately bounded, and that there exist unbounded wedges Wi{i = 1,2,3,4), a constant M > 0, and a continuous function V: [0, oo) x Rn —> [0, oo) such that (i) and (ii) of Theorem 6.6 hold. Moreover, suppose that either

(i) for any (3 > 0 there exists h > 0 such that for any x:R —> Rn with xto G X and JC:|/O,OO) -^Rn being continuous, at any t > to + h with max{\xto\x, suptQ<s<t\x(s)\} < (3

andV(t,D(t,xt)) >MandV(s9x(s)) < qoW2oBoWxx (v(t9D(t9xt))^Jfors G [t-h9t],

we have V(2A)(t9D(t9xtj) < W4(\D(t9xt)\)9

where q: [0, oo) —• [0, oo) is continuous and q(s) > s for s > 0, or (ii) for any a > 0 and (3 > 0 there exists h(a) > 0 such that for any N > 0 and

any x:R —• Rn with xto G X and x: [to, oo) —• Rn being continuous, at any t > to + h with max{\xto\x, $wpto<s<t\x(s)\} < (3 and V(t9D(t9xt)) > M and V(s,x(s)>j < N for s E[t — h91], we have

V(2.»(t9D(t9xt)) < F(t9 V(t9D{t9xt))9N) + a,

where F: [0, oo) x [0, oo) x [0, oo) —» R satisfies the same conditions as those in Theorem 6.6.

Then the solutions of (2.1) are (X9 Rn)-weakly uniformly ultimately bounded.

For illustrative purposes, we now consider the following neutral Volterra integrodif-ferential equations

d r °° pt l - [x(0 - Yfii{*,x(t - rà) ~ J ^ G(t,s,x(s)) ds -f{t)\

(6.3) '=1

= Ax(f) + YiAi(t9x(t - n)) + I H(t9s9x(sj) ds+g(t)9 i=\

where (1) A is an n x n stable constant matrix, and thus there exist a positive definite n x n

matrix P and constants /? > a > 0 so that

ATP + PA = - / , a1xTx < xTPx < (52xTx\

(2) f,g: [0,oo) —> Rn are continuous and there exist constants M\,M2 > 0 so that l/(0| < Mi and \g(t)\ < M2 for f > 0;

(3) BhAr. [0, oo)xRn-> Rn are continuous and Bt(t, 0) = 4</, 0) = 0; (4) G, # : [0, oo)xRxRn —>Rn are continuous and G(Y, 5,0) = //(>, s,0) = 0. Let

(6.4) D(t, ft = <K0) - £ B,(t, <K-r,j) - / G(t, t + s, <f>(s)) ds -f(t)

and V(t,x) = xrFx. It follows that

V(63)(t,D(t,xt)) / OO

- -DT(t,xdD(t,xt) + 2DT(t,xt)P\Y][Ai(t,x(t - n)) + ABi(t,x(t-rl))}

+ f_jH(t,s,x(s)) +AG(t,s,x(s))]ds + g(t)+Af(t)}.

Therefore, if all conditions of Proposition 3.8 hold, then D is (X,i?w)-uniformly stable with Q(e) = ^ | e (if/(f) = 0) and (X, i?w)-uniformly bounded with S(H) = ( 1 + ^ M l . Moreover, suppose that there exist ai > 0 and a2 > 0 such that

(5) for any t > to > 0 and </>Gl,we have

oo

£ SUp k( / ,<Ks) )+^ / (^ ) ) | /=1 -'•/<s<0

+ T |//(f, f0 + s9 <l>(s)) +AG(t910 + s, </>(s)) \ds

<OC\$j)\x.

(6) for any continuous x: [to, oo) —» #w we have

X ; | ^ ( ^ ^ - ^ ) ) + ^ / ( ^ ^ - ^ ) ) | + / | ^ ( f , J ,x (5 ) )+^G(f ,5 ,^ ) ) |&<a 2 sup |*(s)|,

where N is an integer such that r^ < t — to < rx+i. Then

max{KU, sup \x(s)$ < s([V(D(t,xt))}l/2/a)

implies

max{|x,0|x, sup |x(s)|} < ^ - i .

Thus

V{63)(t,D(t,x,)) < -DT(t,Xl)D(t,xt) + 2\D(t,xt)\\P\[a2 sup \x(s)\ t0<s<t

+ ai\xt0\x + M2 + \A\M1]

< -DT(t,x,)D{t,xt)

+ 2\D(t,x,)\\P\

M\ {a.\ + a2) . « ( l - £ i )

+ M2 + \A\Mi

(a{ + a2)\D(t,x,)\

\-k{

fi(ax+a2){\+K) a(\-kx)

= " I l ~ 2\p\^'.',r^x' ^))Dr(t,xt)D(t,xt)

•2\P\[M2 + \A\Mi + ^f^)m,x,)\.

Therefore by Theorem 4.1, we have

PROPOSITION 6.8. Suppose that (l}-(6) and all conditions of Proposition 3.8 hold. Then

(i) ifoc\ + 0C2 < 2(1+^^1' tnen tne s°luti°ns of(6-V are (X, Rn)-uniformfy bounded;

(ii) if ct\ + a2 < 2(i+K)lm andfif) = g(0 = °> then the solutions of (6.3) are (X, Rn)~ uniformly bounded and the zero solution of (6.3) is (X, Rn)-uniformly stable.

Now we consider the (X, #w)-uniformly asymptotic properties of solutions of (6.3) under assumptions (l)-(4). Let all conditions of Proposition 5.7 hold and/(/) = g(t) = 0. Then the D-operator is (X, Rn)-pseudo uniformly asymptotically stable with P(r) — qr. Moreover, suppose that there exists a constant (3\ > 0, for any e > 0 and M > 0 there exists an integer K = K(s,M) > 0 such that

(7) for any x:R—*Rn with xtQ G X,x: [to, oo) —> Rn being continuous and

max{ |x,0 \x, sup \x(s)\} < M9 s>t0

we have

J VK\H(t, t + s,x(t + sj) +AG(t91 + s,x(t + s)) | ds

oo

+ Y, \Mt,x(t - «)) +ABi(t,x(t - n))\ < e i=K+\

and

J \H(t, t + s,x(t + s)) +AG(t, t + s,x(t + s))\ds

IS

+ Y,\Ai(t9x(t - nj) +ABi{t9x(t - n))\ < Pi sup \x(s)\ i=l t-rK<s<t

for/ > to+rjc-

Now, for any M > 0 and b > a > 0, choose e and S > 0 so that

^ ( ' • " T ) ^

and let h = rK,K — K(e,M). Then for any x:R-^ Rn with x,0 G X and x: [to, oo) being continuous, at any t>to+h with

max{|x/o|^, sup |x(s)|} < M, a < K(f,D(f,jCf)) (;,x,)) < -Dr(*,x,)£>(f,x,)

+ 2|Z)(f,x,)| |P| j i : | ^ ( / , x ( / - r,)) + ^ ( / , x ( / - r0)|

+ y |//(/, f + 5, x(> + sj) +AG(t91 + s, x(t + j)) | ds

oo

+ 5] k ^ x ^ - r ^ + ^ / ^ x ^ - r O ) ! i=K+\

+ /" ^ | / / ( / , / + 5,x(^ + 1s))+^G(^,/ + x(r + 5 , ) ) |* |

< -DT(t,xt)D(t,xt) + 2|£>(/,x,)| |P|(/3i sup \x(s)\ + e) t-rK<s<t

< -DT(t,xt)D(t,xt) + 2|P|/3!-^|D(*,x,)|2

orq

•2\D(t,xt)\\P\[t+i3x^}

<--D(t,xt)D{t,xt)-—i 1 2" v """ / " v """" 2/32

a2^ a V & J

<-{\-2\m^)m,x,t. Therefore by Theorem 6.3 and 6.4 we have the following

PROPOSITION 6.9. If all conditions of Proposition 3.8 hold and 4\P\l3\f32 < a2q, then the origin (X, Rn)-attracts solutions of (6.3) uniformly. Moreover, iff(t) andgif) are bounded, then the solutions of (6.3) are (X,Rn)-weakly uniformly ultimately bounded.

ACKNOWLEDGEMENTS. The authors would like to thank the referee for pointing out a mistake in the original manuscript.

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Department of Mathematical Sciences

University of Memphis

Memphis, Tennessee 38152

U.S.A.

Department of Mathematics, Statistics and Computing Science

Dalhousie University

Halifax, Nova Scotia

B3H3J5

Department of Mathematics and Statistics York University North York, Ontario M3J 1P3

Department of Mathematics and Statistics

McMaster University

Hamilton, Ontario

L8S4K1

comparison theorems of liapunov-razumikhin type …ruan/mypapers/haddock... · the generalization...

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